4.5 The Substitution Rule

# 4.5 The Substitution Rule - cos sin u du u C = + ∫ sin...

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The Substitution Rule Section 4.5

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Antidifferentiation of a Composite Function Pattern Recognition: Change of Variables - u- substitution: If u = g(x) then du = g’(x)dx and f(g(x)) g’(x)dx = F(g(x)) + C f(u)du = F(u) + C
Multiplying & Dividing by a Constant Many integrands contain the essential part (the variable part) of g’(x) but are missing a constant multiple. In such cases, you can multiply and divide by the necessary constant multiple.

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Guidelines for Making a Change of Variables 1. Choose a substitution u = g(x) (Choose the inner part.) 3. Compute du = g’(x)dx. 5. Rewrite integral in terms of u. 7. Evaluate the integral in terms of u. 9. Replace u by g(x) . 6. Check answer by differentiating.
The Power Rule for Integration 1 , 1 1 n n u u du C n n + = + +

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Trigonometric Integral Formulas

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Unformatted text preview: cos sin u du u C = + ∫ sin cos u du u C = -+ ∫ 2 sec tan u du u C = + ∫ 2 csc cot u du u C = -+ ∫ sec tan sec u u du u C = + ∫ csc cot csc u u du u C = -+ ∫ Change of Variables for Definite Integrals If the function u = g(x) has a continuous derivative on the closed interval [ a, b ] and f is continuous on the range of g, then a b f(g(x)) g’(x) dx = g(b) g(a) f(u) du Integration of Even and Odd Functions Let f be integrable on the closed interval [ a, -a ]. 1. If f is an even function, then- a a f(x) dx = 2 a f(x) dx 2. If f is an odd function, then - a a f(x) dx = 0 Joke Time How do you catch a unique rabbit? “Unique” up on him! How do you catch a tame rabbit? The “tame” way!...
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## This note was uploaded on 12/01/2011 for the course MATH 112 taught by Professor Jarvis during the Fall '08 term at BYU.

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4.5 The Substitution Rule - cos sin u du u C = + ∫ sin...

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